Journal of Statistical Physics

, Volume 132, Issue 5, pp 863–879 | Cite as

Normal Transport Properties in a Metastable Stationary State for a Classical Particle Coupled to a Non-Ohmic Bath

  • P. Lafitte
  • P. E. Parris
  • S. De Bièvre


We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility μ over macroscopic time scales of order exp (c/F), during which it finds itself in a metastable stationary state. The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties.


Normal transport Inelastic Lorentz gas Diffusion Mobility 


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  1. 1.
    Allinger, J., Weiss, U.: Non-universality of dephasing in quantum transport. Z. Phys. B 98, 289–296 (1995) CrossRefADSGoogle Scholar
  2. 2.
    Bruneau, L., De Bièvre, S.: A Hamiltonian model for linear friction in a homogeneous medium. Commun. Math. Phys. 229, 511 (2002). See also Sect. 6 in mp_arc 01-275 MATHCrossRefADSGoogle Scholar
  3. 3.
    Caldeira, A.O., Leggett, A.J.: Quantum tunnelling in a dissipative system. Ann. Phys. 149, 374 (1983) CrossRefADSGoogle Scholar
  4. 4.
    Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica A 121, 587 (1983) CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Castella, F., Erdös, L., Frommlet, F., Markowich, P.A.: Fokker-Planck equations as scaling limits of reversible quantum systems. J. Stat. Phys. 100, 543 (2000) MATHCrossRefGoogle Scholar
  6. 6.
    Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady-state electrical conduction in the periodic Lorentz gas. Commun. Math. Phys. 154, 569 (1993) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Derivation of Ohm’s law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209 (1993) CrossRefADSGoogle Scholar
  8. 8.
    Cohen, D.: Quantum dissipation versus classical dissipation for generalized Brownian motion. Phys. Rev. Lett. 78(15), 2878–2881 Google Scholar
  9. 9.
    Cohen, D.: Unified Model for the study of diffusion localization and dissipation. Phys. Rev. E 55(2), 1422–1441 (1997) CrossRefADSGoogle Scholar
  10. 10.
    De Bièvre, S., Parris, P., Silvius, A.: Chaotic dynamics of a free particle interacting with a harmonic oscillator. Physica D 208, 96–114 (2005) MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    De Bièvre, S., Lafitte, P., Parris, P.: Normal transport at positive temperatures in classical Hamiltonian systems. In: Proceedings of Transport and Spectral Problems in Quantum Mechanics, Cergy-Pontoise, September 2006. Contemporary Mathematics, Vol. 447, p. 57 (2007) Google Scholar
  12. 12.
    de Smedt, P., Dürr, D., Lebowitz, J.L., Liverani, C.: Quantum system in contact with a thermal environment: rigorous treatment of a simple model. Commun. Math. Phys. 120, 195 (1988) MATHCrossRefADSGoogle Scholar
  13. 13.
    Eckmann, J.P., Young, L.S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262, 237 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6, 504 (1965) MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Grabert, H., Schramm, P., Ingold, G.-L.: Quantum Brownian motion: the functional integral approach. Phys. Rep. 168(3), 115–207 (1988) CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Haki, V., Ambegaokar, V.: Quantum theory of a free particle interacting with a linearly dissipative environment. Phys. Rev. A 32, 423 (1985) CrossRefADSGoogle Scholar
  17. 17.
    Holstein, T.: Studies of polaron motion, part 1: the molecular-crystal model. Ann. Phys. (N.Y.) 281, 706 (1959) CrossRefADSGoogle Scholar
  18. 18.
    Jaksic, V., Pillet, C.A.: Ergodic properties of classical dissipative systems, I. Acta Math. 181, 245 (1998) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kubo, R., Toda, M., Hashitume, N.: Statistical Physics II. Springer, Berlin (1998) MATHGoogle Scholar
  20. 20.
    Larralde, H., Leyvraz, F., Mejia-Monasterio, C.: Transport properties of a modified Lorentz gas. J. Stat. Phys. 113, 197 (2003) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Larralde, H., Leyvraz, F., Mejia-Monasterio, C.: Coupled normal heat and matter transport in a simple model system. Phys. Rev. Lett. 86, 5417 (2001) CrossRefADSGoogle Scholar
  22. 22.
    Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., Zwerger, W.: Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1 (1987) CrossRefADSGoogle Scholar
  23. 23.
    Morgado, R., Oliveira, F., Batrouni, C., Hansen, A.: Relation between anomalous and normal diffusion in systems with memory. Phys. Rev. Lett. 89, 100601 (2002) CrossRefADSGoogle Scholar
  24. 24.
    Schramm, P., Grabert, H.: Low-temperature and long-time anomalies of a damped quantum particle. J. Stat. Phys. 49, 767 (1987) CrossRefADSGoogle Scholar
  25. 25.
    Silvius, A.A., Parris, P.E., De Bièvre, S.: Adiabatic-nonadiabatic transition in the diffusive Hamiltonian dynamics of a classical Holstein polaron. Phys. Rev. B 73, 014304 (2006) CrossRefADSGoogle Scholar
  26. 26.
    Ziman, M.: Electrons and Phonons. Clarendon, Oxford (1962) Google Scholar
  27. 27.
    Zwanzig, R.: Nonlinear generalized Langevin equations. J. Stat. Phys. 9(3), 215–220 (1973) CrossRefADSGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé, CNRS, UMR 8524 et UFR de MathématiquesUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq CedexFrance
  2. 2.Equipe-Projet SIMPAFCentre de Recherche INRIA FutursVilleneuve d’Ascq CedexFrance
  3. 3.Department of PhysicsMissouri University of Science and TechnologyRollaUSA

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